3.61 \(\int \frac{\sqrt{d-c^2 d x^2} (a+b \sin ^{-1}(c x))^2}{f+g x} \, dx\)

Optimal. Leaf size=1442 \[ \text{result too large to display} \]

[Out]

(a^2*Sqrt[d - c^2*d*x^2])/g - (2*b^2*Sqrt[d - c^2*d*x^2])/g - (2*a*b*c*x*Sqrt[d - c^2*d*x^2])/(g*Sqrt[1 - c^2*
x^2]) + (2*a*b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/g - (2*b^2*c*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g*Sqrt[1 - c^
2*x^2]) + (b^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/g + (c*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*g*S
qrt[1 - c^2*x^2]) - ((1 - (c^2*f^2)/g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*(f + g*x)*Sqrt[1 -
c^2*x^2]) + (Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*(f + g*x)) - (a^2*Sqrt[c^2*f^
2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^2*Sqrt[1 - c^2*
x^2]) + ((2*I)*a*b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f -
Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) + (I*b^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*
Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) - ((2*I)*a*b*Sqrt[c^2*f^
2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sq
rt[1 - c^2*x^2]) - (I*b^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*g
)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) + (2*a*b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyL
og[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) + (2*b^2*Sqrt[c^2*f^2 - g^
2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[
1 - c^2*x^2]) - (2*a*b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[
c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) - (2*b^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*PolyLog
[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) + ((2*I)*b^2*Sqrt[c^2*f^2 -
g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^
2]) - ((2*I)*b^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^
2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2])

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Rubi [A]  time = 3.05467, antiderivative size = 1442, normalized size of antiderivative = 1., number of steps used = 38, number of rules used = 23, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.697, Rules used = {4777, 4765, 683, 4757, 4799, 1654, 12, 725, 204, 4797, 4677, 8, 4773, 3323, 2264, 2190, 2279, 2391, 4619, 261, 2531, 2282, 6589} \[ \frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}+\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt{1-c^2 x^2}}-\frac{\left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt{1-c^2 x^2}}+\frac{b^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2}{g}-\frac{2 b^2 c x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g \sqrt{1-c^2 x^2}}+\frac{2 a b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g}-\frac{a^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{f x c^2+g}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^2 \sqrt{1-c^2 x^2}}+\frac{i b^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{1-c^2 x^2}}+\frac{2 i a b \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{1-c^2 x^2}}-\frac{i b^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{1-c^2 x^2}}-\frac{2 i a b \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{1-c^2 x^2}}+\frac{2 b^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{1-c^2 x^2}}+\frac{2 a b \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{1-c^2 x^2}}-\frac{2 b^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{1-c^2 x^2}}-\frac{2 a b \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{1-c^2 x^2}}+\frac{2 i b^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{1-c^2 x^2}}-\frac{2 i b^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{1-c^2 x^2}}-\frac{2 a b c x \sqrt{d-c^2 d x^2}}{g \sqrt{1-c^2 x^2}}+\frac{a^2 \sqrt{d-c^2 d x^2}}{g}-\frac{2 b^2 \sqrt{d-c^2 d x^2}}{g} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(f + g*x),x]

[Out]

(a^2*Sqrt[d - c^2*d*x^2])/g - (2*b^2*Sqrt[d - c^2*d*x^2])/g - (2*a*b*c*x*Sqrt[d - c^2*d*x^2])/(g*Sqrt[1 - c^2*
x^2]) + (2*a*b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/g - (2*b^2*c*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g*Sqrt[1 - c^
2*x^2]) + (b^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/g + (c*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*g*S
qrt[1 - c^2*x^2]) - ((1 - (c^2*f^2)/g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*(f + g*x)*Sqrt[1 -
c^2*x^2]) + (Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c*(f + g*x)) - (a^2*Sqrt[c^2*f^
2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^2*Sqrt[1 - c^2*
x^2]) + ((2*I)*a*b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f -
Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) + (I*b^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*
Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) - ((2*I)*a*b*Sqrt[c^2*f^
2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sq
rt[1 - c^2*x^2]) - (I*b^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*g
)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) + (2*a*b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyL
og[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) + (2*b^2*Sqrt[c^2*f^2 - g^
2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[
1 - c^2*x^2]) - (2*a*b*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[
c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) - (2*b^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*PolyLog
[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2]) + ((2*I)*b^2*Sqrt[c^2*f^2 -
g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[1 - c^2*x^
2]) - ((2*I)*b^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^
2 - g^2])])/(g^2*Sqrt[1 - c^2*x^2])

Rule 4777

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 - c^2*x^2)^FracPart[p], Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a +
b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IntegerQ
[p - 1/2] &&  !GtQ[d, 0]

Rule 4765

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :>
Simp[((f + g*x)^m*(d + e*x^2)*(a + b*ArcSin[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[1/(b*c*Sqrt[d]*(n
+ 1)), Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x], x] /; FreeQ[
{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 4757

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.) + (g_.)*(x_) + (h_.)*(x_)^2)^(p_.))/((d_) + (e_.)*(x_))^2,
x_Symbol] :> With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcSin[c*x])^n, u, x] - Dist
[b*c*n, Int[SimplifyIntegrand[(u*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c
, d, e, f, g, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]

Rule 4799

Int[(ArcSin[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegran
d[(d + e*x^2)^p, RFx*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] &
& IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]

Rule 1654

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 4797

Int[ArcSin[(c_.)*(x_)]^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = ExpandIntegrand[(d + e*
x^2)^p*ArcSin[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] && I
GtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]

Rule 4677

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 4773

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol]
:> Dist[1/(c^(m + 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a,
b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])

Rule 3323

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[((c + d*x)^m*E
^(I*(e + f*x)))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4619

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
(x*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{f+g x} \, dx &=\frac{\sqrt{d-c^2 d x^2} \int \frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{f+g x} \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac{\sqrt{d-c^2 d x^2} \int \frac{\left (-g-2 c^2 f x-c^2 g x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^3}{(f+g x)^2} \, dx}{3 b c \sqrt{1-c^2 x^2}}\\ &=\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt{1-c^2 x^2}}-\frac{\left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt{1-c^2 x^2}}+\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}+\frac{\sqrt{d-c^2 d x^2} \int \frac{\left (\frac{1}{f+g x}-\frac{c^2 \left (g x+\frac{f^2}{f+g x}\right )}{g^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt{1-c^2 x^2}}-\frac{\left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt{1-c^2 x^2}}+\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}+\frac{\sqrt{d-c^2 d x^2} \int \left (-\frac{a^2 \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 (f+g x) \sqrt{1-c^2 x^2}}-\frac{2 a b \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)}{g^2 (f+g x) \sqrt{1-c^2 x^2}}-\frac{b^2 \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)^2}{g^2 (f+g x) \sqrt{1-c^2 x^2}}\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt{1-c^2 x^2}}-\frac{\left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt{1-c^2 x^2}}+\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac{\left (a^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{g^2 \sqrt{1-c^2 x^2}}-\frac{\left (2 a b \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{g^2 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \sin ^{-1}(c x)^2}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{g^2 \sqrt{1-c^2 x^2}}\\ &=\frac{a^2 \sqrt{d-c^2 d x^2}}{g}+\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt{1-c^2 x^2}}-\frac{\left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt{1-c^2 x^2}}+\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac{\left (a^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{c^2 g^2 \left (c^2 f^2-g^2\right )}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{c^2 g^4 \sqrt{1-c^2 x^2}}-\frac{\left (2 a b \sqrt{d-c^2 d x^2}\right ) \int \left (\frac{c^2 g x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\frac{\left (c^2 f^2-g^2\right ) \sin ^{-1}(c x)}{(f+g x) \sqrt{1-c^2 x^2}}\right ) \, dx}{g^2 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 \sqrt{d-c^2 d x^2}\right ) \int \left (\frac{c^2 g x \sin ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}+\frac{\left (c^2 f^2-g^2\right ) \sin ^{-1}(c x)^2}{(f+g x) \sqrt{1-c^2 x^2}}\right ) \, dx}{g^2 \sqrt{1-c^2 x^2}}\\ &=\frac{a^2 \sqrt{d-c^2 d x^2}}{g}+\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt{1-c^2 x^2}}-\frac{\left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt{1-c^2 x^2}}+\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac{\left (2 a b c^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{g \sqrt{1-c^2 x^2}}-\frac{\left (b^2 c^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \sin ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}} \, dx}{g \sqrt{1-c^2 x^2}}-\frac{\left (a^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{g^2 \sqrt{1-c^2 x^2}}-\frac{\left (2 a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \int \frac{\sin ^{-1}(c x)}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{g^2 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \int \frac{\sin ^{-1}(c x)^2}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{g^2 \sqrt{1-c^2 x^2}}\\ &=\frac{a^2 \sqrt{d-c^2 d x^2}}{g}+\frac{2 a b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g}+\frac{b^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2}{g}+\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt{1-c^2 x^2}}-\frac{\left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt{1-c^2 x^2}}+\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac{\left (2 a b c \sqrt{d-c^2 d x^2}\right ) \int 1 \, dx}{g \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 c \sqrt{d-c^2 d x^2}\right ) \int \sin ^{-1}(c x) \, dx}{g \sqrt{1-c^2 x^2}}+\frac{\left (a^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-c^2 f^2+g^2-x^2} \, dx,x,\frac{g+c^2 f x}{\sqrt{1-c^2 x^2}}\right )}{g^2 \sqrt{1-c^2 x^2}}-\frac{\left (2 a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{c f+g \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt{1-c^2 x^2}}\\ &=\frac{a^2 \sqrt{d-c^2 d x^2}}{g}-\frac{2 a b c x \sqrt{d-c^2 d x^2}}{g \sqrt{1-c^2 x^2}}+\frac{2 a b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g}-\frac{2 b^2 c x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g \sqrt{1-c^2 x^2}}+\frac{b^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2}{g}+\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt{1-c^2 x^2}}-\frac{\left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt{1-c^2 x^2}}+\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac{a^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 c^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{g \sqrt{1-c^2 x^2}}-\frac{\left (4 a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x^2}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt{1-c^2 x^2}}\\ &=\frac{a^2 \sqrt{d-c^2 d x^2}}{g}-\frac{2 b^2 \sqrt{d-c^2 d x^2}}{g}-\frac{2 a b c x \sqrt{d-c^2 d x^2}}{g \sqrt{1-c^2 x^2}}+\frac{2 a b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g}-\frac{2 b^2 c x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g \sqrt{1-c^2 x^2}}+\frac{b^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2}{g}+\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt{1-c^2 x^2}}-\frac{\left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt{1-c^2 x^2}}+\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac{a^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{\left (4 i a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c f-2 i e^{i x} g-2 \sqrt{c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{\left (4 i a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c f-2 i e^{i x} g+2 \sqrt{c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{\left (2 i b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x^2}{2 c f-2 i e^{i x} g-2 \sqrt{c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{\left (2 i b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x^2}{2 c f-2 i e^{i x} g+2 \sqrt{c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\\ &=\frac{a^2 \sqrt{d-c^2 d x^2}}{g}-\frac{2 b^2 \sqrt{d-c^2 d x^2}}{g}-\frac{2 a b c x \sqrt{d-c^2 d x^2}}{g \sqrt{1-c^2 x^2}}+\frac{2 a b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g}-\frac{2 b^2 c x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g \sqrt{1-c^2 x^2}}+\frac{b^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2}{g}+\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt{1-c^2 x^2}}-\frac{\left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt{1-c^2 x^2}}+\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac{a^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{2 i a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{i b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{2 i a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{i b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{\left (2 i a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{2 i e^{i x} g}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{\left (2 i a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{2 i e^{i x} g}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{\left (2 i b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-\frac{2 i e^{i x} g}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{\left (2 i b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-\frac{2 i e^{i x} g}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\\ &=\frac{a^2 \sqrt{d-c^2 d x^2}}{g}-\frac{2 b^2 \sqrt{d-c^2 d x^2}}{g}-\frac{2 a b c x \sqrt{d-c^2 d x^2}}{g \sqrt{1-c^2 x^2}}+\frac{2 a b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g}-\frac{2 b^2 c x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g \sqrt{1-c^2 x^2}}+\frac{b^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2}{g}+\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt{1-c^2 x^2}}-\frac{\left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt{1-c^2 x^2}}+\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac{a^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{2 i a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{i b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{2 i a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{i b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{2 b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{2 b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{\left (2 a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i g x}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{\left (2 a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i g x}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (\frac{2 i e^{i x} g}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{\left (2 b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (\frac{2 i e^{i x} g}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\\ &=\frac{a^2 \sqrt{d-c^2 d x^2}}{g}-\frac{2 b^2 \sqrt{d-c^2 d x^2}}{g}-\frac{2 a b c x \sqrt{d-c^2 d x^2}}{g \sqrt{1-c^2 x^2}}+\frac{2 a b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g}-\frac{2 b^2 c x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g \sqrt{1-c^2 x^2}}+\frac{b^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2}{g}+\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt{1-c^2 x^2}}-\frac{\left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt{1-c^2 x^2}}+\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac{a^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{2 i a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{i b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{2 i a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{i b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{2 a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{2 b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{2 a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{2 b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{\left (2 i b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i g x}{c f-\sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{\left (2 i b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i g x}{c f+\sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\\ &=\frac{a^2 \sqrt{d-c^2 d x^2}}{g}-\frac{2 b^2 \sqrt{d-c^2 d x^2}}{g}-\frac{2 a b c x \sqrt{d-c^2 d x^2}}{g \sqrt{1-c^2 x^2}}+\frac{2 a b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g}-\frac{2 b^2 c x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g \sqrt{1-c^2 x^2}}+\frac{b^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2}{g}+\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b g \sqrt{1-c^2 x^2}}-\frac{\left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x) \sqrt{1-c^2 x^2}}+\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c (f+g x)}-\frac{a^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{2 i a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{i b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{2 i a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{i b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{2 a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{2 b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{2 a b (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{2 b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{2 i b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \text{Li}_3\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{2 i b^2 (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \text{Li}_3\left (\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 1.44346, size = 516, normalized size = 0.36 \[ \frac{\sqrt{d-c^2 d x^2} \left (3 b c (f+g x) \left (i \sqrt{c^2 f^2-g^2} \left (-2 i b \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )+2 i b \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )+2 b^2 \text{PolyLog}\left (3,\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )-2 b^2 \text{PolyLog}\left (3,\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )+\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right )-\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )\right )+g \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-2 b g \left (a c x+b \sqrt{1-c^2 x^2}+b c x \sin ^{-1}(c x)\right )\right )+\left (c^2 f^2-g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^3+c^2 g x (f+g x) \left (a+b \sin ^{-1}(c x)\right )^3+g^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^3\right )}{3 b c g^2 \sqrt{1-c^2 x^2} (f+g x)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(f + g*x),x]

[Out]

(Sqrt[d - c^2*d*x^2]*((c^2*f^2 - g^2)*(a + b*ArcSin[c*x])^3 + c^2*g*x*(f + g*x)*(a + b*ArcSin[c*x])^3 + g^2*(1
- c^2*x^2)*(a + b*ArcSin[c*x])^3 + 3*b*c*(f + g*x)*(g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*b*g*(a*c*x
+ b*Sqrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x]) + I*Sqrt[c^2*f^2 - g^2]*((a + b*ArcSin[c*x])^2*Log[1 + (I*E^(I*ArcS
in[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - (a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqr
t[c^2*f^2 - g^2])] - (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]
)] + (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] + 2*b^2*PolyL
og[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] - 2*b^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + S
qrt[c^2*f^2 - g^2])]))))/(3*b*c*g^2*(f + g*x)*Sqrt[1 - c^2*x^2])

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Maple [F]  time = 0.549, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}}{gx+f}\sqrt{-{c}^{2}d{x}^{2}+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/(g*x+f),x)

[Out]

int((a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/(g*x+f),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/(g*x+f),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )}}{g x + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/(g*x+f),x, algorithm="fricas")

[Out]

integral(sqrt(-c^2*d*x^2 + d)*(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)/(g*x + f), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}{f + g x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x))**2*(-c**2*d*x**2+d)**(1/2)/(g*x+f),x)

[Out]

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*asin(c*x))**2/(f + g*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{g x + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/(g*x+f),x, algorithm="giac")

[Out]

integrate(sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)^2/(g*x + f), x)